ISNM66: International Series of Numerical Mathematics Internationale Schriftenreihe zur Numerischen Mathematik Serie internationale d' Analyse numerique Vol.66

Edited by Ch. Blanc, Lausanne; A. Ghizzetti, Roma; R. Glowinski, Paris; G. Golub, Stanford; P. Henrici, Ziirich; H.O. Kreiss, Pasadena; A. Ostrowski, Montagnola; J. Todd, Pasadena

Birkhauser Verlag Basel · Boston . Stuttgart

Numerical Solution of Partial Differential Equations: Theory, Tools and Case Studies Summer Seminar Series Held at CSIR, Pretoria, February 8-10, 1~

1983

Edited by D.P.Lauri~

Birkhauser Verlag Basel·· Boston· Stuttgart

Addresses of the contributors

Mn. M. L Baart Dr. T. Gevecl Dr. D. P. Laurie Dr. J.M. Navon Dr. H. Nelshlo• Mrs. H . .A.. Rlphagen National Research Institute for Mathematical Sciences CSIR P.O. Box39S Pretoria (0001 South Africa)

Dr. B. M. Herbst Dr. S. W. Schoomble Department of Applied Mathematics University of the Orange Free State P.O. Box339 Bloemfontein (9300 South Africa)

Library of Congre .. Cataloging in Publication Data

Numerical solution of partial differential equations. (International series of numerical mathematics ;

v. 66 = Internationale Schriflenreihe zur Numerischen Mathematik ; v.66)

Includes bibliographical references. I. Differential equations, Partial--Numerical

solutions--Congresses. I. Laurie, D.P. (Dirk Pieter), 1946- Il. Series: International series of numerical mathematics ; v. 66. QA374.N86 1983 SIS.3'53 83-21480 ISBN 3-7643-1561-X

CIP-Kurzlitelaufnahme tier Deutschen Bibliothek

Numerical •olulion of partial differential equations: theory, tools and case studies : proceedings of a summer seminar series held at CSIR, Pretoria,8-10 February 1982 I ed. by D. P. Laurie. - Basel ; Boston ; Stuttgart : Birkhluser, 1983.

(International series of numerical mathematics ; Vol. 66) ISBN 3-7643-1561-X

NE: Laurie, Dirk P. [Hrsg.]; South African Council for Scientific and Industrial Research; GT

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form of by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

® 1983 Birkhlluser Verlag Basel Printed in Switzerland by Birkhlluser AG, Graphisches Untemehmen, Basel ISBN 3-7643-1561-X

5

CONTENTS

PREFACE

PART 1: THEORETICAL BACKGROUND

CHAPTER 1 \_ __

SOME TYPES OF PARTIAL DIFFERENTIAL EQUATIONS AND ASSOCIATED WELL-POSED PROBLEMS

(T. Geveci)

1. INTRODUCTION

2. NOTATION AND SOME FUNCTION SPACES

3. ELLIPTIC BOUNDARY VALUE PROBLEMS

4. TIME-DEPENDENT EQUATIONS

REFERENCES

CHAPTER 2

BASIC PRINCIPLES OF DISCRETIZATION METHODS (D.P.Laurie)

1. INTRODUCTION

2. FINITE DIFFERENCE METHODS

3. FINITE ELEMENT METHODS

4. COMPARISON OF THE FINITE ELEMENT AND FINITE DIFFERENCE METHODS

REFERENCES

10

13

14

17

32

50

52

53

64

73

74

6.

CHAPTER 3

CONVERGENCE OF DISCRETIZATION METHODS

(T~Geveci)

1. INTRODUCTION

2. AN EXAMPLE

3. ANALYSIS OF THE CONVERGENCE OF THE. FI.NITE ELEMENT METHOD FOR ELLIPTIC BOUNDARY-VALUE PROBLEMS

4. THE FINITE DIFFERENCE ANALYSIS OF THE CAUCHY PROBLEM

5. A GALERKIN TYPE APPROXIMATION SCHEME FOR .THE HEAT EQUATION WITH THE DIRICHLET BOUNDARY CONDITION

6. GALERKIN TYPE APPROXIMATION SCH.EMES FOR THE WAVE EQUATION AND SOME GENERAL COMMENTS

REFERENCES

CHAPTER 4

BASIS FUNCTIONS IN THE FINITE ELEMENT METHOD

(L.Baart)

0. INTRODUCTION

1. THE SPACE OF ADMISSIBLE FUNCTIONS

2. THE GENERAL INTERPOLATION PROBLEM, APPROXIMATION

3, BASIS FUNCTIONS IN ONE DIMENSION

4, BASIS FUNCTIONS FOR TWO-DIMENSIONAL RECTANGULAR AND POLYGONAL REGIONS

5. BASIS FUNCTIONS FOR TWO-DIMENSIONAL REGIONS WITH CURVED BOUNDARIES

6. NONCONFORMING ELEMENTS. THE PATCH TEST

76

78

93

101

118

127

131

134

135

137

139

143

14B

161

7

7. FINITE ELEMENTS IN THREE DIMENSIONS

8. CONCLUSION

REFERENtES---

CHAPTER 5

TIME DISCRETIZATION IN PARABOLIC EQUATIONS

(D.P.Laurie)

1. INTRODUCTION

2. THEORETICAL SOLUTION

3. TWO-LEVEL METHODS

4. THREE-LEVEL METHODS

5. AUTOMATIC STEP SIZE CONTROL

6. IMPLEMENTATION OF RATIONAL APPROXIMATIONS

7. SPLITTING METHODS

8, CONCLUSION

REFERENCES

CHAPlER 6

PARABOLIC EQUATIONS WITH DOMINATING CONVECTION TERMS

(B.M. Herbst and S,W. Schoombie)

1. INTRODUCTION

2. APPROXIMATION ON A NON-UNIFORM GRID

3. NON-UNIFORM GRIDS AND TIME-DEPENDENT EQUATIONS

4. COMPUTATIONAL CONSIDERATIONS

REFERENCES

163

164

165

167

168

171

175

176

177

179

181

181

185

186

1B7

191

201

8

CHAPTER 7

SOLUTION OF LARGE SYSTEMS OF LINEAR EQUATlONS

(D.P. Laurie)

1. INTRODUCTION

2. DIRECT METHODS

3. ITERATIVE METHODS

4. THE CONJUGATE GRADIENT METHOD

5. PRECONDITIONED CONJUGATE GRADIENT METHODS

6. COMPARISON OF METHODS

REFERENCES

PART 2 CASE STUDIES

CHAPTER 8

FINITE-ELEMENTS MESH PARTITIONING FOR TIME INTEGRATION OF TRANSIENT PROBLEMS

(H.Neishlos)

1. INTRODUCTION

2. GOVERNING EQUATIONS

3. TIME INTEGRATION

4. MESH PARTITIONING

5. REMARKS

REFERENCES

204

206

210

213

216

220

222

225

226

228

231

243

243

9

CHAPTER 9

NUMERICAL WEATHER PREDICTION

(H.A. Riphagen)

1. INTRODUCTION

2. MODEL EQUATIONS FOR ATMOSPHERIC MOTIONS

3. NUMERICAL METHODS

4. A SIMPLE FILTERED MODEL

REFERENCES

CHAPTER 10

CONSERVATION LAWS IN FLUID DYNAMICS AND THE ENFORCEMENT OF THEIR PRESERVATION IN NUMERICAL DISCRETlZATlONS

(l.M.Navon)

0. INTRODUCTION

1. THE THEORETICAL APPROACH TO CONSERVATION LAWS

2. DISCRETE APPROXIMATIONS OF CONSERVATION LAWS

3. 'A POSTERIORI' METHODS FOR ENFORCING CONSERVATION LAWS IN DISCRETIZED FLUID-DYNAMICS EQUATIONS

4. INTEGRAL CONSTRAINTS FOR THE TRUNCATED SPECTRAL EQUATION

5. CONSERVATION LAWS AND FINITE-ELEMENT DISCRETIZATION

REFERENCES

246

247

253

261

274

286

288

306

326

327

330

334

286

CHAPTER 10

CONSERVATION LAWS IN FLUID DYNAMICS AND THE ENFORCEMENT

OF THEIR PRESERVATION IN NUMERICAL DISCRETIZATIONS

I. M. Navan

l. INTRODUCTION

The existence of conservation laws for fluid dynamics pl'oblems has been

recognized to be of vital importance to the understanding of basic physi=

cal and mathematical properties of problems of interest. Almost all the

equations describing fluid dynamics problems can be interpreted as the

laws of conservation of mass, momentum and energy. In this short lecture

note we will adopt two different approaches which essentially cha.racterize

different lines of thought as far as conservation laws are concerned.

The first is the 'theoretical' approach whereby one considers what is

a conservation law and the problem of obtaining conservation laws in the

sense of physics from a system of equations as an integrability problem

on a manifold (Eiseman and Stone 1980 [lJ).

In this approach we are also concerned with the preservation of conserva=

tion laws under coordinate transformations. One then might ask how many

conservation laws are contained in a system of partial differential equa=

tions describing some fluid dynamic phenomenon. This brings us to the

Korteweg-de Vries equations which possess an infinity of conservation laws

and to other similar equations. This can be connected with .other types

of conservation ·laws recently found for the KdV equation by Wahlquist

and Estabrook [2]. For another excellent survey on this topic see Manin

[3]. This topic is also related to the inverse scattering problem.

287

For the discretized non-linear partial differential equations of fluid

dynamics the author has preferred to bring examples from numerical weather ' prediction and the accent was put mainly on finite difference schemes.

Here conservative spatial finite difference discretizations . were frequently

used in order to reduce the tendency towards non-linear instability

(~ary [4], Kalnay de Rivas et al [5]).

There is still a controversy if the use of a conservative spatial approxi=

mation will totally avoid non-linear instability (Gary [4J). However in

atmospheric sciences and numerical weather prediction it is coillnon ex=

perience that enstrophy conserving schemes and quadratically or energy

conserving schemes provide a more accurate approximation to the spectrum

and avoid the unbounded growth associated with catastrophic non-linear

instability.

It should be stressed that in any numerical model, the 6~n-l.te 4e.6o.tut.ion

.impo.&e.6 an Mti6.i.cia.t 'unll' a..t the .&ho4t-unve end· 06 the .6pedltum, in=

ducing an excessive accumulation of energy in the shortest waves (Kalnay

de Rivas et al [5]). This problem is worst in non-conservative schemes

but it appears even in alias-free, energy and enstrophy conserving spectral

models. This justifies using besides the conservative finite-difference

schemes some parametrization of the unresolved subgrid eddies to withdraw

energy from the smallest resolved scales (Kalnay de Rivas et al [5]),

(Gary [4J).

In this lecture-note we have stressed only some aspects of finite diffe=

rence conserving schemes. A survey of 'a posteriori' methods for enforcing

conservation in discretized finite difference models is also provided.

Two small sections are dedicated to the conservative properties of . spectral

and finite element discretization methods.

288

Finally it should be mentioned that we are far from having exhausted the

fluid-dynamics conservation laws (see Mobbs (6)).

We have also not treated systems of non-linear ·hyperbolic partial differen=

tial equations in conservation law form (see Lax [7J), the entropy cona i=

tion, shock-waves, rarefaction-waves and contact discontinuities. The

interested reader is referred to Liu [8J • . [9J, Glium, [10] and Glium

and Lax [llJ.

1. THE THEORETICAL APPROACH TO CONSERVATION LAWS

1.1 The Basic Equations

Ve6-i.nltlon 1. A conservative law on a manifold is defined exactly as

any exact differential 1-form e whose image h6 under a linear operator

h, is also exact.

Ve6~n<.,ti.on 2. The rate of change of a quantity U in a volume M is classi=

cally expressed as the negative sum of fluxes F(U) of a quantity U across

a boundary aM.

~t f UdV = f F(U)dS M

where dV and dS are volume' elements on M and aM respectively. Using

(1)

Stokes theorem and the fac:t that M was arbitrarily° chosen, we obtain finally

the partial differential equation

au + 'ii • F(U) = o at

in cli.ve.JtgelUCe 6oJtm or in conservation law form.

Ex.ample 1

(2)

The divergence form of a heat conducting viscous 'fluid is given by the

continuity equation

289

*+'i/· (pV) = 0, (3)

the energy equation

~+ 'i/ •JEV - K'i/T + [-rJ·V} = 0 (4)

K thermal conductivity E total energy

p density v velocity vector

T temperature [T] the stress tensor

a the tensor product,

and the momentum equation

a ar(pV) + 'i/ • {pV a v + [T)}= 0 (5)

(6) •

where E is the internal energy, and E the total energy . The stress-tensor

for a Newtonian fluid is:

[T] = (p ~ :W•V)[IJ • 2µ[0] (7)

where

p is pressure

A,µ viscosity coefficients

[I] the identity dyadic

[0) the deformation tensor

For an ideal gas the equation

where

of state is

p = pRT

cp specific heat at constant pressure

R the gas constant

E internal energy = CvT

a is the tensor product

given by:

(8)

CV specific heat at constant volume

R = Cp - CV

290

The conservation law fonn of the fluid dynamic equations is important for

the numerical computation of certain flow fields. This is particularly

true for flows with shock waves where in fact the Rankine-Hugoniot condi=

tions can be satisfied by a· mesh adjustment.

1.2 Fluid dynamic equations in curvilinear coordinates

Following the approach of Eiseman and Stone [lJ let x1 , ... ,xn be fixed

Cartesian coordinates and let y1 , •.. ,yn be curvilinear coordinates. The

local vector fields {a/ax1 , ••• ,a/axn} fonn the basis of Rn while

{a/ay1 , ••. ,a/ayn} are an alternate basis related to the first by

(9)

using the Einstein sunmation convention.

We assume the Jacobian J = det(axi/ayj) to be different from D and we

denote

ei = a/ayi • i=l,2, .. . ,n. (10)

A metric on the vector fields is an _inner product < ·, ·>. By applying

the metric to the basis elements ei we get

axa axa a a gi. =< e.,e. >= :-,- < - , 3 >

J 1 J ay ai axa ay

axa axa axa axa = - - cS = - -, summation on a.

ay1 ayJ aa ay1 ayJ (11)

The differential element of arc length ds can be obtained from the rela=

tionship: a a

(ds) 2 = cS 0

dxa dxa = cS 0 ~ ~ dyw ay0 • g, __ d-:,fl dy0• (12)

o., ap ayw ayo ~

If A is the Jacobian transfonnation axi/ayj we have gij = AAt; then

g ~ det(gij) = det AAt = (det A) 2 • J 2 (13)

291

i.e. (gij) is non-singular if J;O i.e. the matrix gij has an inverse which

will be denoted by (gij).

Following Eiseman and Stone [lJ, we note that to express the system (3)-(5)

of fluid dynamic equations in curvilinear coordinates we need to define

a dual basis {e\ .. ,l!n} by

< Lej >= o~ 1 1

(14)

where

(15)

Then the gradient is defined as_

'V = ek 3 Dk (16)

and the divergence 'V is defined by replacing the tensor product in 'V by

a dot product. To obtain the dot product one applies the· metric to ek

and the first component of the tensor on the right.

Furthennore if IQ= IJI the absolute value of the Jacobian is independent

of time, each equation can be multiplied by lg and lg can also be brought

under the time derivative.

The continuity equation then becomes

a a · at(plg") + --,(pv1/9) = D •

ay

The energy equation becomes

}i-(E/9) + 4(Evi-gijK aTJ + g . Tij vr)/gJ = O ay ay rJ

and the momentum equation is given by:

where

(17)

(18)

(19)

292

_i gja agai agak agi k l":k = --,.- { ---,.;-- + ~ - - } , ~ ay~ ay, aya

(20)

is the Levi-Civita tensor, designed to be compatible with the metric

(21)

while

(22)

In the curvilinear coordinate system the continuity and energy equations

remain in conservation law form wh.lie .the momentum equ.a.tlon m.l.16e.o th.-i.6

6oJun due to .the 6oU/lee-.ii.h.e teJun sir r{r· To transform it into a censer=

vation law form one has to rewrite the momentum equations in a way that

absorbs the source-like term. By using the work of Anderson, Preiser

and Rubin [12] and implementing the method of integrating factors, one

obtains momentum equations in conservative form:

~t(pvs lg axm) +_a_ (Bij ~) = 0 m=l,2, .. . n . (23) axs ai' ay1

The conservation law form of the fluid dynamic equations is important for

the numerical solution of flow fields. In particular this is true for

flows with shock-waves ·where the Rankine-Hugoniot conditions can be sa=

tisfied by a mesh-adjustment.

1.3 Conservation law forms from Stoke's theorem and differential forms

In a region M of an M-dimensional Euclidean space a quantity U is con=

served as a function of time if the rate of change of U in M is equal to the

negative sum of the flux w of quantity U across the bounda~y aM. Usi~g

Flanders [13] notation of exterior differential forms, the conservation

of U is given by

293

~t f U 0 dV = - f w M oM

(24)

for a positively oriented n-form dV as a volume element on M and for some

(n-1) form w which is some function of U that describes the U-flux through

aM. If x1 ,x2, .•• ,xn are Cartesian coordinates ordered in such a way that

dV = dx1 11 dx 2

11 ••• 11 dxn the flux can be expressed as

n w = r (-1) 1 F1 (U) 0 dx1

11 •• • 11 dxi-l 11 dxi+l 11 •• • 11 dxn i=l

(25)

where Fi is the flux in direction xi.

When U is a vector then Fi are vectors in the same space and w is a (n-1)

vector form. Using Stokes theorem we obtain

au a n aFi f at 0 dV = at f U 0 dV = f. w = f dw = f ( r ~) 0 dV M M oM M M i=lax,

(26)

Remembering that the region M is arbitrary, we obtain the conservation

law form

au · n ~-at + E 1 - O •

i=l ax (27)

An alternative definition of a conservation law is any pde in n indepen=

dent variables {x1 , ... ,xn} .and p dependent variables {y1, ... ,yP} which

has the form:

n a1/Ji E ~= 0

i=l ax

where 1jJi are functions of y1 , •.• yP.

hample 2

(28)

The equations describing an inviscid ideal fluid in general curvilinear

coordinates are given by the system

(29)

where

and

294

r P = (y-1) [E - W: {1L} 2 )p]

r P

aj - are contravariant coefficients

y = cp/cv

1.4 Expression of a system of pde's in~enns of conservation laws

Consider the conversion of a system of the fonn

a a ···· at(u) + (h) ax (u) s o

where (u) is a column vector

(h) is a square matrix with entries depending only on (u) .

(30)

(31)

(32)

(33)

Can we find a suitable nonsingular matrix depending only on (u) to obtain

a system of conservation laws.

(34)

where U and V are column-vectors depending only on (u)?

Example 3

Let us take the equations of one dimensional time-dependent, nonisentropic,

adiabatic fluid flow:

2!15

= 0 (35)

p, u, p, y - are density, velocity, pressure and adiabatic constant .res=

pectively.

If S is entropy and p=p(p,S) and one assumes y > 1 then (35) is equivalent

to the system

where p = yp/p = c2 > O. p

(36)

Note that the eigenvalues of the 3x3 matrices which appear in (35) or

(36) are distinct. These are u-c, u, u+c, and their corresponding eigen=

vectors are (O,pc2 ,-c),(c2 ,0,l) and (O,pc2 ,c). If p > 0, which is the

physical case, we can premultiply the system (35) by the nonsingular

matrix

(37)

to obtain the following system of conservation laws

[

pu j pu 2+p = 0 .

pu3+ 2y pu Y-T

(38)

296

1.5 Conservation laws and variational principles (Noether's theorem)

Let us take again the form n a,1,i i: =... = 0

i=l ax1 (39)

The functions wi may themselves be in the form of derivatives . Let f be . a j .

a function of yJ, ~and x1, j=l, •. . ,p and i~l, ... ,n.

ax Let

Y~. - ¥rj i - , X = [X ]

1 ax (40)

We consider the n-fold integral f fdx where R is a closed bounded region R

and we assume that the integral is invariant under a family of transfor=

mations with an arbitrary finite number of parameters. Further we suppose

that f does not explicitly depend on then independent variables xi which

can then be taken as parameters.

If we now look for a stationary value for f fdx we obtain p Euler varia= R

tional equations, which are:

n a fyj 2 i: ~,fi = O .

i=l ax Yi (41)

"Noether's theorem is a means of associating a conservation equation with

an infinitesimal transformation having an action integral invarfant.

In our notation Noether's · result asserts that n linear combinations of

(41) which have the form of conservation laws can be found. I.e., there

exist functions w~. 0 ,; k ,; n such that: 1

n a awk l: ~ f al=:-,+

i=l ax 1 Yi ax

aw~ ... n " o

ax (42)

where a~ is an (nxp) matrix and the range of summation on a is from 1 to

p.

297

Other relations with similar form are

a · a · ~ (ylk) + :-k (-y~) = 0 ax 1 ax 1

(43)

The number of conservation laws which appear in (43) depends in general

on the group of transformations which fixes f fdx. R

Example 4

We seek a stationary value for

J = f f ( w2 + w2 )dx dy ( 44) R x . y

where R is the unit disc. Here we have n=2, p=l. The conservation laws

we can obtain from (44) have the form

(45)

The first of the above relationships is the variational equation

wxx + wyy = t.w = O • ( 46)

One observes that J is invariant under a one-parameter transformation

group given by

That is

x• = x(cos e) + y(sin e) (47)

y• = -x(sin e) + y(cos e)

f f (w; + w~)dx dy = f f (w;. + w~.Jdx* dy* where R = R* . R R* (48)

When the number of dependent variabl es is greater than two, the problem

of obtaining conservation laws becomes more difficult. See Osborn [15].

For the existence of conservation laws on manifolds see Eiseman and Stone [1

298

fXlllllp,f.e. 5

Conservative fonn of the time-dependent Navier-Stokes equations

The Navier-Stokes equations for plane-flows are written in cartesian

coordinates (x,y) as follows.

au + aF + aG = 0 at ax ay

The unicolumn matrice U,F and G have the following expressions:

U = [p,pu,pv,pEJT

while F and Gare split into

or component-wise

inviscid tenns viscous tenns (R,S)

F = Fl (U) - R(W,oW/ax,aW/ay)

G = G1 (U) - S(W,aW/ax,aWjay)

H • [:J p = (y-l)pe

lpv l puv

pv2+p

(pE+p)v

(49)

(50)

(51)

(52)

(53)

(54)

(55)

0

R = 'xy

299

s =

0

'yx='xy

'yy

ur:xx + vr:xy + c: ~~ K - thennal conductivity

ur: + vr: + ...! ae xy yy cv ay

q ·- KgradT

r: = (>.+2µ) 2!:!_ + ). ~ xx ax ay Stress tensor r: _ (au + av)

xy - µ ay ax r: = ). 2-!!. + (A+2µ) av yy ax ay

).,µ - viscosity coefficients

). : ).(T), µ =µ (T), K=K(T)

3). + 2µ ~ 0, µ ~ 0

(56)

(57)

(58)

If one assumes local thermodynamk equilibrium one always makes use of the

Stokes relation

3). + 2µ = 0 (59)

See also Viviand [16J and Vinokut [17].

1.6 The Korteweg de Vries Equation and its Conservation Laws

We consider the equation

(60)

and its generalizations.

In this context a conservation law associated with an equation such as

(60) is expressed by an equation of the fonn

(61)

300

where T the con6e1r.ved den6Lty and -X, the flux of T, are functionals of

u.

De.6.i.n.Ui.on. (Kruskal, Muira and Gardner [18J)

Local conservation laws

If Tis a loca.l functional of u, i.e. if the value of Tat any x depends

only on the values of u in an arbitrarily small neighbourhood of x, then

Tis a l.oca.l con6e11.ved dl'.n.6,i;ty, If Xis also local, then (61) is a

local con6e11.va.tlon laM. If T is in particular a polynomial in u and its

x derivatives are not dependent explicidy on x and t, then we call T a

polynomial conserved density; if X is also such a polynat1ial we call (61)

a polynomial conservation law. There .is a close relationship between

constants of motion and conservation laws.

For KdV equations (for a polynomial conservation law) T and X are each a0 a1 ai.

a finite sum of terms of the form u0

, u1 , .. . ,ui. where

and aj are nonnegative integers.

_ aju uj - axJ (62)

The rank is defined as the sum of the number of factors aj and half the

number of x differentiations

2. r1 z i: (1 + iJ)a . •

j=O J (63)

A polynomial of rank r is one whose terms are all of rank r.

There is a polynomial conservation law with nontrivial conserved density

Tr for each posi.tive integral rank r (and corresponding flux Xr of rank

r+l).

Historically it is to be noted that Korteweg and de Vries Cl9J chose to

301

exhibit their equation in conservation law fonn i .e.

an - 1 .& a (! n2 + i an + ~ a a2n2l at - 2 2. ax ax (64)

n surface elevation above equilibrium level

a small arbitrarily constant related to the uniform motion of the liquid

g gravitational constant

a

p liquid density

T surface capillary tension

The KdV equation describes the evolution of long water waves down a canal

of rectangular cross-section. The KdV equation can be rewritten as

(65)

by the rescaling transformations

t' = ir;I t , X' - Ta , u -ln-~a (66)

where we have dropped the primes.

The KdV equation is Galilean invariant. (Muira [20J). Conservation laws

can be used for deriving 'a priori' estimates and to obtain integrals of

the motion. Three first conservation laws for the KdV equation. are

(67b)

(67c)

The first is the KdV equation in conservation law ·fonn. The second i s

obtained by multiplying· by 2u and the third by multiplying by 3u2-uxx and

algebraically manipulating the differentiations.

302

These conserved densities can be interpreted as mass, momentum and ·ener\l.)t

conservations for some physical systems. The infinity of conserved densi~

ties beyond these first three do not seem to have any physical interpre~

tation. The proof of the existence of an infinite member of polynomial

conservation laws was given by Kruskal, Muira, Gardner and Zabusky [18],

Gardner [21] and Kruskal [22J. Muira [23] studied the polynomial censer=

vation laws of a general class of KdV equations

(68)

This was also the starting point for the inverse scattering method for

the exact solution of the KdV equations.

1.7 Conservation laws for generalized KdV eguations

If we consider the class of equations

(69)

(with all coefficients equal to 1, witnout loss of generality) Kruskal

and Muira [22] obtained the following results:

a) if r is even (Burger's equation p=l, r=2, then there exists only one

polynomial conservation law i.e. the equation itself)

b) If r is odd i.e. r=2q+l, q=O,l there exist always three conservation

laws for any p ;,; 0 with the conserved densities given by

(70)

For q=O, we define a new variable v = uP+l to get the equation

303

(71)

which always has infinitely many conservation laws.

For the case p=O, the equation is linear and there are infinitely many

conservation laws with conserved densities

T = (anu/axn) 2 n=0,1,2, •.. (72)

The cases p=l and p=2 with q=l correspond to the KdV equation or the modi=

fied KdV equation respectively. All the other cases have only the three

(polynomial) conservation laws.

0 2 p "' 3

0 00 00 00 00

1 00 00 00

3

q "' 2 I 00 3 3 3

The results and concepts developed for the KdV equation like conservation

laws have been extended. to other nonlinear pde's like the nonlinear

Schrodinger equation

(73)

and the sine-Gordon equation (see Lamb [24J)

uxt = sin u (74)

or

utt - uxx + sin u = O • (75)

304

1.8 Benney' s equations and their conservation laws

These are fully nonlinear long-wave equations describing the motion over

a flat bottom of a two dimensionally inviscid fluid with a free surface

in a gravitational field in the long-wave approximation.

The equations have the form

y

subject to boundary conditions

ht + u0 hx - v0 = 0 )

y = p P

0 = constant

and v = O at y = 0.

(76.1)

(76 .2)

(76.3)

h(x,t) (76.4)

The flat rigid bottom is denoted by y = 0 and the free surface by y=h(x,t)

(Benney [25J). Here u and v are the horizontal and vertical components

of velocity, p is density.

- --- -

305

[f one defines the following integrals

h(x,t) A (x,t) = f u(x,y,t)ndy, n=0,1,2, • .. (A

0(x,t) = h(x,t))(77)

n o

it was shown by Benney [25J that this system of equations possesses an

infinite number of conservation laws of the form

Tt + Xx = 0 (78)

where T the conserved density and -X, the flux, are polynomials in the

An n=0,1,2,... . Muira [23J proved that the fully nonlinear long-wave

equations possess an infinite number of local conservation laws in the

form

(79)

where T the local conserved densities and -X,-Y the loca l fluxes are

polynomials in n·and the An.

Eqn (76.1) is already in conservation form. Multiplying (76.1) by u and

adding to (76.2) yields a second conservation law

(80)

To find the third conservation law multiply (76.2) by u giving

(81)

Using (76 . l} to replace vy by -ux and integrating the last term by parts

and again using (76.1) we obtain

(82)

T.hese three conservation laws have the usual interpretation of conserva=

tion of mass, momentum and energy respectively .

306

Benney (25) obtained the fo11ow.ing result for these equations: If u(x,y,t)

and h(x,t) are continuous functions with continuous first-order partial

derivatives for - 00 < x < + m • 0 s y s h, and satisfy (76.1)-(76.4) then

(83)

For other extensions see Muira [26J.

2. DISCRETE APPROXIMATIONS OF CONSERVATION LAWS

2.1 The Arakawa approach for an 'a priori' method for enforcing conserva=

tion laws on the discrete approximation of a simple flu id-dynamics

equation

We shall introduce Arakawa's method [27] by considering the vorticity

equation

·~ 2 at+ v.v~ = o, ~ = v "' (84)

where the velocity V is assumed to be nondivergent i.e. it can be ex=

pressed as

(85)

where ljJ is a stream function and the vorticity is

(86)

i.e.

(87)

and substituting (86) into (84) . We obtain

a 2 a 2 2 ~ V 1/J =at~= J(V ljl,ljl ) = J(~,ljl) = -J(ljl,V 1/J), (88)

This equation gives the local change in vorticity as a result of advection

by a two-dimensional nondivergent velocity.

307

If the Gauss divergence theorem is applied to (84) over a closed regi on

and there is no flux at the boundaries or the region is closed, the right-

hand side of (84) vanishes. This means

-~ = 0 at (89)

where ~ means the integral over the region, so that we obtain that the

total (or mean) vorticity is conserved. If (84) is multiplied by ~ we

obtain

(90)

Application of the Gauss divergence theorem again shows that the total

or mean square vorticity called e.n6-tlt.ophy is also conserved .

Finally, multiplying (SB) by ljl, we obtain

(91)

Modifying each term as follows

(92)

ljJV·(~V) = V·(ljl~V) - ~V·Vlji (93)

and substituting these expressions in (91) gives

Vljl•V ~ = V· (ijJV ¥tJ + V· (ljl~V) - ~V·VijJ • (94)

The term -~V·Vljl vanishes identically since

V .l to Vljl . (95)

Also

(96)

where Vi/J•Vljl i s twice the kinetic energy. When the Gauss theorem is applie

308

the rightchand side of (94} vanishes and we obtain

a(''71ji·Vljl}/ at • o (97)

which means that the mean k.lne.tic. eneJ!Sy is .conserved over a closed

region. This means that for the vorticity equation (84) over a closed

region, mean kinetic energy, mean vorticity and enstrophy are conserved.

In a famous paper published in 1959, using the vorticity equatio.n, Phillips

[28J showed that the leapfrog centred space-differencing scheme leads to

nonlinear instability with a catastrophic 'blow-up' in kinetic energy in

the short wavelenghts range. This is contrary to the results of Fj~rtoft

(29) who has shown that, in the continuous case, total kinetic energy

is conserved and also that there is no cascade of energy towards short

wavelengths in the continuous case, i.e. the average unve numbeJt. .<.,o al.ho

C!Onl>eJt.Ved.

Phillips (28) found that a mere reduction in grid-size or time increment

did not eliminate the nonlinear instability. The instability could how=

ever be controlled by periodically removing wavelengths 4d and smaller.

This was done either by the application of a space filter

(98)

or by an artificial diffusing term such as Dv2A, where ·o is the eddy­

diffusivity, or finally by using a selective finite-difference scheme.

All these methods are not very satisfying as they distort the solution

by affecting longer waves in the process of a long-term numerica~ inte=

gration.

Arakawa [27J sought to prevent a false cascade of energy and enstrophy

by maintaining the conservation of mean enstrophy, kinetic energy and

vorticity for the barotropic vorticity equation for a time-continuous space

- ~-~ -

309

difference discretization. He started with the barotropic vorticity equ<

tion written in Jacobian form

a~ ) at = - J(ljl.~ (99)

by considering each of the products

~J(ljl, ~ ) and ljlJ(ljl,~) (100 :

with several different finite difference Jacobians to determine whether

they conserve kinetic energy or enstrophy. ·

The essence of the Arakawa 'a priori' method will be exposed now.

The analytic Jacobian can be expressed as

( 101 )

Using the following 9 point stenci l

6 2 5

EE: we can obtain the following finite difference Jacobians

(102)

(103}

(104}

The superscript + or x denotes the points from which the finite differenc

approximations for the derivatives of 1jl and ~ (in that order) are formed

310

A + symbol indicates use of the points (1,2 ,3 ,4) and the x symbol ind i=

cates the use of points (5,6,7,8). All the Jacobians (102-104) are of

the general bil-inear fonn

-1,0,+l Jij(ip,i;;} = k,.e.:m,n ci,j,k,R.,m,n ljli+k,j+.e. i;;i+m,j+n • (105)

The coefficients c can be determined so that the finite difference di s=

cretization possesses as many of the conservation properties of the ana•

lytic (continuous) differential equation as desired. If (99) is multi=

plied by i;;ij the left-band side represents .the local rate of change of 2

l;ij'

In this case the right-hand side may then be written as

-1,0,l E a .. i j 1;. j 1; •• m,n l ,J, +m, +n i+m, +n i ,J ·

(106)

The rate of change of 1;2 at the point (i,j) may be considered to be the

consequence of interaction between the given point and adjacent points.

Also the change of 1;2 at point (i+m,j+n) will receive a corresponding con=

tribution from point 1;ij" A net change of mean square vorticity (i.e.

non-conservation) may be avoided if the gain at the point (i,j) due to

interaction with point (i+m,j+n) is exactly balanced by a corresponding

loss at (i+m,j+n) due to point (i,j). This occurs if the coefficients

a fulfil the condition

ai ,j, i+m,j+n = -a i+m ,j+n, i ,j (107)

Consider now the gain in square vorticity at point o due to the value at

point 1 and vice versa. The results, excluding the term 4d 2 are

(108)

(109)

- --- ----- - -

311

Since the tenns do not cancel and no other opportunity for the product

1;0

1;1

occurs over the grid, the sum i;;J++ cannot vanish in general and ~2

will not be conserved i.e.

Similarly

+x l;o Jo - 1;0 1;5 (ip1-ip2) + . .. (110)

1;5 J+x -5 1;5 1;0(1j12-ljl1) + •.. (lll)

and

l; Jx+ - -0 0 1;0 1;1 (ips-ipa) + ... (112)

1;1 Jx+ -1 1;1 1;o(ip2-ip'+) + . .. (113)

Arakawa (27J found that J+x does conserve c2 since the contributions at

adjacent points exactly cancel one another, while a comparison of terms

comprising i;;J++ and 1;Jx+ shows that they are opposite in sign. So it is

clear that

(114)

A similar examination of the finite difference approximation of ljlJ( ip , 1; )

shows that

Jx+ + i (J++ + J+x) conserves K. (115)

Sy combining (114) and (115) it is seen that the Arakawa Jacobian

(116)

conserves enstrophy ~ 2 , mean vorticity~ and mean kinetic energy R Arakawa [27) has shown that his Jacobian also conserves the me.an IA.tlve

11W11bVL. As a result of these discrete conservation properties, the Arakaw

312

Jacobian prevents the continued growth of very short waves, which is charac=

teristic of ~onlinear instability. Aliasing is still present in the form

of phase errors but these also result from linear finite-difference trun=

cation-errors. Lil.ly [30] has given a rigorous proof by spectral methods

that Arakawa's quadratic conserving scheme eliminates the type of nonlinear

instability demonstrated by Phillips [28]. The conservative properties

of the Arakawa Jacobian have been established for the time continuous case,

but not for time discretization. For instance when the time derivative

is approximated with the leapfrog time differencing scheme and the Arakawa

Jacobian is used, it follows that

(117)

Hence

so,n so,n+l = so,n so,n-1 • (118)

But this is not the correct conservation property for mean square vort i=

city, which should be written as

s~.n+l = ~~.n = s~.n-1 ' etc. (119)

If the time differencing gives

~o,n = (so,n+l + so,n-1)/2 (120)

the above (119) property holds but th is requires a rather complicated

implic.it scheme. We shall come back to this shortcoming in another sec=

tion. It has been demonstrated by Jespersen [31J that the Arakawa Jacobian

is a special case of a finite-element method using rectangular ·9-noded

elements.

313

2.2 Generalization of Arakawa's scheme to the shallow-water equations

(Gramme 1 tvedt [32J)

We write the equations for a barotropic, frictionless, divergent flow

with a free surface (shallow-water equations) as

~ + u ~ + v ~ - fv + a¥' = o at ax ay . ax (12la)

av + u av + v av + fu + a.p = 0 at ax ay ay (12lb)

(121c)

The velocities u and v are in the x and y directions respectively and

¥>=gh is the geopotential of the fluid where g is the acceleration of

gravity and h is the height of the free surface; f is the Coriolis para=

meter.

These equations conserve the integrals of mass, total energy and enstroph

in the channel 0 s x ~ O , 0 s y ~ L, t ~ 0. In the following we will

use a regular grid with horizontal spacing llx = lly = ll, and time increment

llt.

To derive the discretized finite-difference equations we will use the

Shuman sum and difference operators

(122)

(123)

(124)

--x . 2 Jxx = (~. x) = a + Tt; "xx l[ (x +•)~(x •)+2a(x )J ~ = • (l i '-' ~ i-u i • (125)

A second and fourth-order finite-difference approximation to the first anc

314

second derivatives of a can then be written as

(126)

(127)

(128)

a~ ' - • i a - i a2x2x + 0(6 ) . ax2 ' xx (129)

Now the Arakawa scheme for the two-dimensional vorticity equation describing

frictionless, non-divergent flow in a closed domain is written in the

Shuman notation

(130)

If we now use the vorticity components u and v in the x and y direction

respectively, instead of the stream function ~ and the vorticity ~. by

using the following relationships

u = - ~~ and v = ~ (131)

i. e.

u ~ - ~y and v = ~x (132)

and if we use second-order finite-difference approximations and ~ = vx-uy

for the vorticity, Grammeltvedt [32J after some lengthy algebra derived

the following generalization of Arakawa's scheme for the barotropic equa=

tions (shallow-water equations).

315

+ 2vXY i1 + vx /} - fvXY + "' = 0 ( 133a) y y x --x y

{2uxy vx + uY v + (vY vY) + i(v(v+n2V2v)) + fuxy + 'l'y 0(133b x x y y

and

~tt + (~x u) + (~Yv) = 0 (133c) x y

where

'2u = u + u xx yy

and the locations of u,v and"' in the grid are given by

u u ~

v b v V'

"' II "' v v

v 0 Vb V'

II II v ·v

"' "'

(134)

Fig.2

For the non-divergent flow, the generalized Arakawa scheme will conserve

mean kinetic. energy, mean vorticity and enstrophy.

2.3 Two general conservation laws for shal low-water equations

If we consider the equations describing the motion in a homogeneous in=

compressible fluid with a free surface having bottom topography, we obtain

ii'.+ <1.!5. x V* + 'il·(K+~) " 0 (135)

~ + 'il·V* = 0 (136)

Here q is the potential vorticity and V* the mass flux defined by

316

q = (f+~}/h (137)

v• = hV (138)

where V is the horizontal velocity, t the time, f the Coriolis parameter,

~ the vorticity, ~=k·VXV , !. is the vertical unit vector, V the horizontal

del operator, h the vertical extent of a fluid column above the bottom

surface, . K the kinetic energy per unit mass, iV2 , g the gravitation

acceleration and hs the bottom surface height and

(139)

If we multiply (135) by V* and combine the results with (136),we obtain

the equation for the time change of total kinetic energy

~t(hK} + V· (V"K) + V*·V<ll = 0 , (140)

If we multiply (136) by <II we obtain the equation for the time change of

potential energy,

(141)

Summation of (140) and (141) yields the equation of the conservation of

total energy

(142)

where the overbar denotes the mean over a finite domain with no inflow

or outflow through the boundaries. The vorticity equation for this fluid

motion is obtained from (135) and can be written as

ft.(hq) + V· (V*q) = 0. (143)

Subtracting (136) times q from (143) and dividing by h gives

317

2.9_ + V·"q = 0 at ' • (144)

Now hq times (144) plus lq2 times (136) gives the equation for the time

change of potential enstrophy (Ertel's theorem [33)).

(145)

which leads to the conservation-law of potential enstrophy

a--2-ar< lhq ) = D. (146)

Arakawa and Lamb [34) derived, after a very lengthy algebra, a potential

enstrophy and energy conserving scheme for the shallow-water (S-W) equa=

tions with bottom topography for general flow. They have shown that their

scheme suppresses the spurious energy cascade towards short-wave numbers

and that the overall flow regime is.adequately represented. This potentia·

enstrophy 'a priori' conserving scheme was generalized by Arakawa and

Lamb to a spherical grid while K.Takano [34] of UCLA derived a fourth-order

accurate version of the scheme, both for square and spherical grids.

2.4 The global barotropic (S-W) equations and their conservation laws

(spherical coordinates)

These equations can be written as:

(147a)

av l a 2 2 at + Z</IU + a ae<"' + Hu +v ) ) 0 (147b)

~ l a a _ at+ .a cos e <aA (<Pu) + 1i6 (<Pv cos e)} - o (147c)

where

z = {f + a c~s 0 (~~ - fe.<u cos e))l/<P (148)

318

is the potential vorticity.

a - radius of the globe

e - longitude

A - latitude

This system has the following conservation laws

conservation of 'mass'

a " 2" 2 a f f "' cos e a dhd0 = 0 t 0 0

!_ J1" __ l_ {av - !. (u cos e)} cos e a2dhde = o at 0 0

a cos e aA ae ·

conservation of vorticity

a" 2" 2 22 2 n;f f {;.+"'Hu +v )}a cos e dhde = o

0 0

conservation of energy

a " 2" 2 2 TI: f f t "' a cos e dhd0 a 0 0 0

conservation of potential enstrophy.

(149)

(150)

(151)

(152)

Sadourny (35] has shown that fonnal conservation of potential enstrophy is

more important than fonnal conservation of total energy, in that potential

enstrophy conserving models of the S-W equations are inherently more stable

and maintain more realistic energy spectra.

In this case we choose to conserve mass. vorticity and potential enstrophy.

Away from the poles such a finite element discretization is (Burridge

[36J)

(153)

319

(154)

""' 1 at = - acose (\ u + o8(v cos 0)) (155)

where the zonal and meridianal mass fluxes are given by

u = ~Au, V • ~9v respectively (156)

and 4

e = l(u2 + co! 9 (v2 cos 9) (kinetic energy) (157)

-.a z = A 9A{f cos 0 + oAv - o9(u cos 0)}

"' cos 0 (158)

and oh, 08 are the two-point · centred derivatives acting on i11111ediate

neighbours along 9 or A directions. These finite differences used with

the regular staggered grid illustrated in fig 2 conserves total mass,

vorticity and the potential enstrophy, apart from boundary fluxes at or

near the poles.

Although energy is not conserved formally by these finite difference fonns

it turns out that the model conserves energy very accurately. Near the

poles a special form of finite differences is required to enforce the

same conservation laws. Details can be found in Burridge (36].

2.5 Conservation laws for baroclinic primitive equations models

The equations for a dry adiabatic version of a baroclinic model are in

the normalized vertical pressure coordinate o

(160)

320

a~ = -RT (hydrostatic equation) (162)

where the vertical coordinate is a and is defined by

a = p/ps a = da/dt (163)

where Ps is the surface pressure.

At the surface p=ps hence a=l and o=O; at the top of the atmosphere

p=O hence a=O and cr=O.

1 1 av a z = Ps f + acose [TI - as (a cos e)J (164)

(165)

(166)

Here

T - i s the temperature

R - gas constant for air

cp- specific heat at constant pressure

The boundary conditions are

(167)

(168)

('no flux' conditions).

For this baroclinic model we require mass and total energy conservation.

Integration of the continuity equation gives

321

(169)

I The divergence tenn in (169) integrates to zero over the globe which means

that the total mass is conserved.

A 'kinetic' energy equation can be derived from the momentum equations

in the fonn

(170)

Using the continuity equation (159), (170) can be written as

In order to construct the total energy equation the pressure gradient

terms must be expressed in another .form. Using the continuity equation

(159) we have

Combining (171) and (172) we have

(173)

Now

(174)

Addition of the kinetic energy equation and the flux form of the thermo=

dynamic equation and integration from a=O to 1 gives

- - - -- .

322

(175)

This equation expresses the energy conservation for the a system (159)­

(163). The divergence term integrates to zero over the whole globe - i.e.

we have that the total energy, potential plus kinetic, is conserved.

2.6 The discretized version of the baroclinic primitive equations which

maintains conservation of energy and mass (Burridge [36J)

The vertical grid (see Fig.3) is such that the primary variables V and T

are carried at integral levels, while the vertical velocity a and the

geopotential 'P are carried at half integer levels. The variable vertical

grid spacing 6ok is defined by

Vertical grid

Index

------------- 0=0,op; o=o~=O

---------------------------------- V,T,w; o=o1

------------- a,ip; a:a3 2

2 ----------------------------------

k ---------------------------------- V,T,w; o=ok

k+~ ' '· :

N ---------------------------------- V,T,w; o=oN

N+!

(176)

!

323

2.6.1 I~~-~g~~~iQ~_Qf_£Q~!i~Yi~~

The vertically discretized continuity equation takes the form

ap 6(p a) ~ + Vo•(psVk) + ti~k k = 0 .

Multiplying (177) by 6ok and summing (instead of integration) from

k=l,2, .•• ,k gives

(as Ps~~ = O) .

For k=N we have

a~ N N ....-- = - E v ·(p Vk)tiok = - v • r v ·(p Vk)tiok •• k=l a s a k•l a s .

(177)

(178)

(179)

This equation is the finite-difference analogue of . (169) which obviously

conserves the total mass of the model atmosphere.

The vertically discretized finite-difference momentum equation is

(180)

where

(181)

The finite difference kinetic energy equation is

324

aek {pscrk+tvk+t"vk-psok-tvk·vk-1l Ps at+ t llak

ll(p O->k • t Vk•Vk l>a~ + PsVk·Vaek + PsVk•{Va"'k+R\Vl.nps} = o. (182)

This equation. may be rewritten in flux form by using the ·finite difference

continuity equation

where

For the hydrostatic equation· we write

l>aop (X'"lncr)k • -RT k

and we obtain a finite-difference analogue of (172)

PsVkVaopk = Va·{psVkopk} - opkVa·(psVk)

. aps lla(Psa)k = Va·{psVkopk} + opk{ at+ l> }

ap ak s . lla(op(a at+ psa))

= Va·{psVkopk} + l>a k

-(llaop)k •Ps • •Ps · ~ [Sk(a at+ Ps0 lk+t + ak(a at+ Ps0 >k-lJ

ak + sk • 1

as

(183)

(184)

(185)

(186)

(187)

325

If we require total energy conservation then the fonn of the last term in

(186) is a constraint in our choice of the w term

(188)

in the thermodynamic equation whose discretized form is

ark ok+t(\+l-Tk)+ok-t(\-Tk-1) Ps at+ PsVk·VaTk + Ps H lla }

k

_ _l [RTwJ = 0 cp a k ' (189)

If the w-tenn is chosen so as to maintain energy conservation we have

(190)

The term [llfn:illa]k can be interpreted as a definition of (l/ak) for the

baroclinic model.

Our remaining degree of freedom is in the choice of the weights ak and Sk

subject to ak+Sk=l. The ECMWF operational model uses ak=Sk=t.

More about this topic can be found in an article by Arakawa and Lamb [37]

describing the UCLA general circulation model.

Mesinger [38] developed a series of enstrophy and energy conserving fin i te­

difference schemes for the horizontal advection. An explicit potential

vorticity conservation by finite-difference schemes framed in generalized

vertical coordinates was developed by Bleck [39J .

326

3. 'A POSTERIORI' METHODS FOR ENFORCING CONSERVATION LAWS IN DISCRETIZED

FLUID-DYNAMICS EQUATIONS

All the 'a priori' numerical schemes modelling the discrete conservation

laws result in rather complicated finite-difference schemes that are

difficult to generalize to fluid-dynamics problems of interest.

A different approach is to enforce the conservation relationships expli=

citly by modifying the forecast field values, at each time step of the

numerical integration.Sasaki ([40J, [41J, [42J) proposed such a variational

approach and applied it to conserve total energy and mass in one and

two-dimensional shallow-water equations models on a rotating plane.

Bayliss and Isaacson [43J, Isaacson and Turkel [44) and Isaacson [45J

presented a simple method of making any finite difference scheme conser=

vative with respect to any qua.ntity. In their approach the conservative

constraints were linearized about the predicted values by means of a

gradient method for modifying the predicted values at each time step of

the numerical integration.

Isaacson et al [46J and Isaacson et al [47) have implemented the same

technique in terms of simultaneous conservation for the shallow-water

equations over a sphere, taking into account orography effects • .

Their approach has been tested by Kalnay-Rivas et al [48J with enstrophy

as the conserved quantity. Ka·lnay-Rivas [48), [49J found that the use

of an enstrophy conserving scheme can be successfully replaced by using

a fourth-order quadratically conserving scheme on a global domain, com=

bined with the periodic application of a 16th-order Shapiro filter re=

moving waves shorter than four times the grid size before they attain

finite amplitude.

327

Navon [50] has shown that it is possible to achieve stable long-term

integrations of the shallow-water equations using the above mentioned

techniques. In these lecture notes a modified Sasaki variational approach

to enforce conservation of total mass and potential enstrophy will be

developed and then a modified Bayliss-Isaacson technique also designed

to enforce conservation of potential enstrophy and total mass, will be

described.

A new approach based on viewing the problem of enforcing conservation

laws in discretized models as a nonlinearly constrained optimization

problem solved by an augmented Lagrangian penalty method will be men=

tioned (Navon-de Villiers [51]).

3.1 Formulation of the modified Sasaki variational method for enforcing

conservation of potential enstrophy and mass

3.2 The numerical variational algorittun

3.3 The Bayliss-Isaacson algorithm

[Ec:lU01t.'.i 110,te: As this material has already been published elsewhere

([50], §2~§4), copyright restrictions do not allow its repetition. It

should nevertheless be regarded as an integral part of the present survey,

and for that reason the paragraph headings have been retained.]

4. INTEGRAL CONSTRAINTS FOR THE TRUNCATED SPECTRAL EQUATION

We shall illustrate in brief the properties of the spectral method by

using the barotropic nondivergent equation

~~ = -V-V(~+f)

with the usual meaning for all symbols.

(191)

328

For a non-divergent flow on the spherical globe we obtain

~ = !•V x V = V2$ (192)

where $ is a scalar stream function. We write

V = (U,V) (193)

where

u = cos 6 ~ and V = .!. ~ · a aip a a:1. (194)

Equation (191) is written in spherical polar coordinates as

(195)

where ip,:I., a and n denote respectively, latitude, longitude and the · earth 's

radius and rotation rate.

Using an expansion of $ in orthogonal surface spherical hannonics, a

truncated expression for the stream function is

(196)

(197)

where p~(sin <P) is the associated Legendre polynomial of the first kind

nonnalized to unity, and the truncation parameter J denotes a rhomboidal

wave number truncation, while m denotes a planetary wave number and 1-m

denotes a meridional wave number.

The nonlinear advection term is written as

and

c • -=--"-­a2 cos <P

~ ~ -

(198)

329

C~ = t E EE E {12 (12+1)-11 (11+1)}$~1$~2L(11 ,12 ,1;m1 ,m2 ,m). (199) m1 \ m2 12 1 2

Here

(200)

Lorenz [58J proved that as a consequence of the non-aliased truncation,

the integral invariants of the non-divergent barotropic vorticity equation

(191) are also valid for the solution to the truncated spectral equa=

tions. Lorenz [58J proved this for plane geometry using a representa=

tion in terms of double Fourier series, while Platzman [59J did the proof

for spherical geometry. Merilees [60] explicitly illustrated the result

that the truncated set of spectral equations retains as invariant the

domain integral of kinetic energy via conservative redistribution of energy

within each individual interaction. For an extensive review on this

issue see Machenhauer [61J).

While spectral approximations conserve the gross characteristics of the

energy spectrum and a systematic energy cascade towards higher wavenumbers

is not possible, a certain btoc.IUng of energy in the highest wavenumbers

occurs. In actual numerical integrations the integral constraints are

of course only approximately valid due to time truncation and round-off

errors.

For three dimensional models the property of the spectral approximation

of non-aliased truncation of the non-linear terms implies quasi-conser=

vation of energy in adiabatic, friction-free numerical integrations.

(Merilees [60J). Weigle [62] has made a detailed study of the conserva=

tion properties of a spectral shallow-water equations model while for

more general sigma-levels models Bourke [63J and Hoskins and Sirrmons [64)

found that energy is very nearly conserved during adiabatic, friction-free

integration.

- -

330

The blocking phenomenon resulting from neglecting interactions involving

components outside the truncation limits was observed by Gordon and Stern

[65J and Bourke [63J.

In long-term low resolution spectrally truncated models the amplitudes

of small-scale components will grow unrealistically large and a scale­

selective damping parametrization is required.

5. CONSERVATION LAWS AND FINITE-ELEMENT DISCRETIZATION

Here we will only refer to the Galerkin finite-element method. This

method (see Cullen [66J) has a property of satisfying certain.conserva=

tion laws.

If we set

w = au ax (201)

and express w = twnXn(x,y) where Xn are basis functions, the Galerkin

method gives

f (w - ~)X = 0 0 ax n (202)

using the basic functions as test functions.

If we multiply the n-th member of the system (202) by un and sum over n

we get

so that

exactly.

f(w - ~)u = O ax

(203)

(204)

In many problems, this equation expresses conservation of energy or mo=

mentum.

331

In general, any conservation law which can be expressed by multiplying

the differential equation by one variable and integrating will be satis=

fied exactly in the Galerkin method. Raymond and Garder [67] however

find that conservation is a disadvantage on irregular grids.

The two-stage highly accurate Galerkin method does not conserve 'energy'

(Cullen [68J). It is likely that this is because the aliasing effect

is removed in this finite element integration and that instability is

less likely. Lee et al [69J considered the Boussinesq inviscid problem

by finite-element discretization and found out that a method which con=

served the quadratic quantities was most stable in terms of time integra=

tion, whereas the standard advective form, which does not conserve any

of the three quantities conserved by the Boussinesq equation namely total

energy (E), total temperature (T) and total temperature squared (T2),

performed very poorly in time integrations.

5. 1 Conservation properties of the Boussinesq equation for finite-elemen

discretization

The equations for an inviscid Boussinesq fluid in a two dimensional regio

o, with boundary r are:

au p [at + .!!_·li'.!!.J = -li'p - PiYT in 0 (205a)

li'•u = O in o (205b)

aT at+ .!!_•li'T = 0 in O (205c)

where p is density, .!!. the velocity, p the pressure, g the acceleration

due to gravity,ythe volumetric coefficient of thermal expansion and T

the temperature.

For the case of a contained flow the boundary conditions are

332

!!"!! = o on r (206)

where!! is the outward pointing unit normal vector on r .

From (205c) we have

~ T" + u·VTn = o at - (207)

so that

c& f Tn = f if = - f !!•VT" = - f V• (!!_Tn) = - f (!!•!!)Tn = O. (208) fl fl fl fl r

Thus any integral of T is conserved.

From (205a) we find

(209)

so

Now

(210)

where.!:. is the position vector.

From (206),(208),(209) and (210) we get:

(211)

This equation expresses the conservation-law of the total energy of the

fluid .

In a breakthrough paper Cliffe [70] found a particular Galerkin discreti=

zation (using two parameters a and B and different finite-element inter=

333

polation spaces for pressure and temperature) such that the total energy

E, the total temperature f and the total temperature squared (f2) were

conserved. He used an eight-noded quadrilateral serendipity element

for the pressure field, while for the temperature field a standard four­

noded element was used. The weights were a=S=i·

5.2 The effect of time-discretization and other errors on conservation

laws (Cliffe [70), Sanz-Serna [71J).

We will first examine the effect of time-discretization. A semi-discrete

equation can be written as

y = f(y) . (212)

To preserve the discrete conservation law we require a non-dissipative

method of integration for (212). The simplest is the trapezoidal rule

(Crank-Nicolson) leading to

(213)

Lee et al [69Jas well as Sanz-Serna [71J have observed that whilst (213)

wi 11 preserve the conservation properties of 1 inear quantities, for

quadratic quantities which are conserved in the semi-discrete equation

(213) the conservation properties are lost.

A second-order non-dissipative method, the midpoint rule, does however

preserve the conservation properties of quadratic quantities i .e.

(214)

Other sources of error are the solution of nonlinear systems of equa=

tions. In exact arithmetic, linear conservation properties are preserved

at each iteration while for quadratic quantities, conservation properties

are only preserved in the limit of convergence.

. - ---- - ~

334

If the region is not regular then .isoparametric transfonnations of the

boundary finite-elements lead to integrals which have to be evaluated

by Gaussian integration schemes introducing quadtuLtu!Le eJULo/UJ. Finally,

all conservation properties are affected by rounding-errors, which are

usually quite small.

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